International
Tables for Crystallography Volume A Spacegroup symmetry Edited by Th. Hahn © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. A, ch. 9.1, pp. 742749
doi: 10.1107/97809553602060000517 Chapter 9.1. Bases, lattices, Bravais lattices and other classifications^{a}Universität Erlangen–Nürnberg, RobertKochStrasse 4a, D91080 Uttenreuth, Germany, and ^{b}Institut für Angewandte Physik, Lehrstuhl für Kristallographie und Strukturphysik, Universität Erlangen–Nürnberg, Bismarckstrasse 10, D91054 Erlangen, Germany In this chapter, the symmetry and some topological aspects of lattices are considered. Different classifications of lattices into five Voronoi types, 24 Symmetrische Sorten and 14 Bravais lattices are described. Conventional lattice bases and their cells are defined and illustrated. Extensive tabulations for the two and threedimensional Bravais lattices are provided. Further classifications use reduction theory; details of the Delaunay reduction are given. 
In three dimensions, a coordinate system is defined by an origin and a basis consisting of three noncoplanar vectors. The lengths a, b, c of the basis vectors a, b, c and the intervector angles , , are called the metric parameters. In n dimensions, the lengths are designated and the angles , where .
Another description of the basis consists of the scalar products of all pairs of basis vectors. The set of these scalar products obeys the rules of covariant tensors of the second rank (see Section 5.1.3 ). The scalar products may be written in the form of a matrix which is called the matrix of the metric coefficients or the metric tensor.
The change from one basis to another is described by a transformation matrix P. The transformation of the old basis (a, b, c) to the new basis is given by The relation holds for the metric tensors G and .
A threedimensional lattice can be visualized best as an infinite periodic array of points, which are the termini of the vectors The parallelepiped determined by the basis vectors a, b, c is called a (primitive) unit cell of the lattice (cf. Section 8.1.4 ), a, b and c are a primitive basis of the lattice. The number of possible lattice bases is infinite.
For the investigation of the properties of lattices, appropriate bases are required. In order to select suitable bases (see below), transformations may be necessary (Section 5.1.3 ). Of the several properties of lattices, only symmetry and some topological aspects are considered in this chapter. Some further properties of lattices are given in Chapter 9.3 .
The treatment of the topological properties is restricted here to the consideration of the neighbourhood of a lattice point. For this purpose, the domain of influence (Wirkungsbereich, Dirichlet domain, Voronoi domain, Wigner–Seitz cell) (Delaunay, 1933) is introduced. The domain of a particular lattice point consists of all points in space that are closer to this lattice point than to any other lattice point or at most equidistant to it. To construct the domain, the selected lattice point is connected to all other lattice points. The set of planes perpendicular to these connecting lines and passing through their midpoints contains the boundary planes of the domain of influence, which is thus a convex polyhedron. (Niggli and Delaunay used the term `domain of influence' for the interior of the convex polyhedron only.) Without the use of metrical properties, Minkowski (1897) proved that the maximal number of boundary planes resulting from this construction is equal to , where n is the dimension of the space. The minimal number of boundary planes is 2n. Each face of the polyhedron represents a lattice vector. Thus, the topological, metrical and symmetry properties of infinite lattices can be discussed with the aid of a finite polyhedron, namely the domain of influence (cf. Burzlaff & Zimmermann, 1977).
Different procedures are in use to select specific bases of lattices. The reduction procedures employ metrical properties to develop a sequence of basis transformations which lead to a reduced basis and reduced cell (see Chapter 9.3 ).
Another possibility is to make use of the symmetry properties of lattices. This procedure, with the aid of standardization rules, leads to the conventional crystallographic basis and cell. In addition to translational symmetry, a lattice possesses pointgroup symmetry. No crystal can have higher pointgroup symmetry than the point group of its lattice, which is called holohedry. The seven point groups of lattices in three dimensions and the four in two dimensions form the basis for the classification of lattices (Table 9.1.4.1). It may be shown by an algebraic approach (Burckhardt, 1966) or a topological one (Delaunay, 1933) that the arrangement of the symmetry elements with respect to the lattice vectors is not arbitrary but well determined. Taking as basis vectors lattice vectors along important symmetry directions and choosing the origin in a lattice point simplifies the description of the lattice symmetry operations (cf. Chapter 12.1 ). Note that such a basis is not necessarily a (primitive) basis of the lattice (see below). The choice of a basis controlled by symmetry is not always unique; in the monoclinic system, for example, one vector can be taken parallel to the symmetry direction but the other two vectors, perpendicular to it, are not uniquely determined by symmetry.
^{†}The symbols for crystal families were adopted by the International Union of Crystallography in 1985; cf. de Wolff et al. (1985).

The choice of conventions for standardizing the setting of a lattice depends on the purpose for which it is used. The several sets of conventions rest on two conflicting principles: symmetry considerations and metric considerations. The following rules (i) to (vii) defining a conventional basis are taken from Donnay (1943; Donnay & Ondik, 1973); they deal with the conventions based on symmetry:
The metric parameters of the conventional basis are called lattice parameters. For the purpose of identification, additional metric rules have to be employed to make the labelling unique; they can be found in the introduction to Crystal Data (Donnay & Ondik, 1973).
When the above rules have been applied, it may occur that not all lattice points can be described by integral coordinates. In such cases, the unit cell contains two, three or four lattice points. The additional points may be regarded as centrings of the conventional cell. They have simple rational coordinates. For a conventional basis, the number of lattice points per cell is 1, 2, 3 or 4 (see Tables 9.1.7.1 and 9.1.7.2).
In two dimensions, only two centring types are needed:
In three dimensions, the following centring types are used: [see rule (iiib) above].
In orthorhombic and monoclinic lattices, some differently centred cells can be transformed into each other without violating the symmetry conditions for the choice of the basis vectors. In these cases, the different centred cells belong to the same centring mode. In the orthorhombic case, the three types of onefacecentred cells belong to the same centring mode because the symbol of the cell depends on the labelling of the basis vectors; C is usually preferred to A and B in the standard setting; the centring mode is designated S (seitenflächenzentriert). In the monoclinic case (bunique setting), A, I and C can be transformed into each other without changing the symmetry direction. C is used for the standard setting (cf. Section 2.2.3 ); it represents the centring mode S. The vectors a, c are conventionally chosen as short as the Ccentring allows so that they need not be the shortest two vectors in their net plane and need not fulfil the inequalities (9.1.4.1).
In some situations, the Icentring of the monoclinic conventional cell may be more advantageous. If the vectors a, c are the shortest ones leading to the centring I, they obey the inequalities (9.1.4.1).
By means of the abovementioned lattice properties, it is possible to classify lattices according to various criteria. Lattices can be subdivided with respect to their topological types of domains, resulting in two classes in two dimensions and five classes in three dimensions. They are called Voronoi types (see Table 9.1.6.1). If the classification involves topological and symmetry properties of the domains, 24 Symmetrische Sorten (Delaunay, 1933) are obtained in three dimensions and 5 in two dimensions. Other classifications consider either the centring type or the point group of the lattice.

The most important classification takes into account both the lattice pointgroup symmetry and the centring mode (Bravais, 1866). The resulting classes are called Bravais types of lattices or, for short, Bravais lattices. Two lattices belong to the same Bravais type if and only if they coincide both in their pointgroup symmetry and in the centring mode of their conventional cells. The Bravais lattice characterizes the translational subgroup of a space group. The number of Bravais lattices is 1 in one dimension, 5 in two dimensions, 14 in three dimensions and 64 in four dimensions. The Bravais lattices may be derived by topological (Delaunay, 1933) or algebraic procedures (Burckhardt, 1966; Neubüser et al., 1971). It can be shown (Wondratschek et al., 1971) that `all Bravais types of the same crystal family can be obtained from each other by the process of centring'. As a consequence, different Bravais types of the same [crystal] family (cf. Section 8.1.4 ) differ in their centring mode. Thus, the Bravais types may be described by a lowercase letter designating the crystal family and an uppercase letter designating the centring mode. The relations between the point groups of the lattices and the crystal families are shown in Table 9.1.4.1. Since the hexagonal and rhombohedral Bravais types belong to the same crystal family, the rhombohedral lattice is described by hR, h indicating the family and R the centring type. This nomenclature was adopted for the 1969 reprint (IT 1969) of IT (1952) and for Structure Reports since 1975 (cf. Trotter, 1975).
In Fig. 9.1.7.1, conventional cells for the 14 threedimensional Bravais lattices are illustrated.
In Tables 9.1.7.1 and 9.1.7.2, the two and threedimensional Bravais lattices are described in detail. For each entry, the tables contain conditions that must be fulfilled by the lattice parameters and the metric tensor. These conditions are given with respect to two different basis systems, first the conventional basis related to symmetry, second a special primitive basis (see below). In columns 2 and 3, basis vectors not required by symmetry to be of the same length are designated by different letters. Columns 4 and 5 contain the metric tensors for the two related bases. Column 6 shows the relations between the components of the two tensors.


The last columns of Tables 9.1.7.1 and 9.1.7.2 show parallel projections of the appropriate conventional unit cells. Among the different possible choices of the primitive basis, as discussed in Sections 9.1.1–9.1.5, the special primitive basis mentioned above is obtained according to the following rules:
If a primitive basis is chosen according to these rules, basis vectors of the conventional cell have parallel facediagonal or bodydiagonal orientation with respect to the basis vectors of the primitive cell. For cubic and rhombohedral lattices, the primitive basis vectors are selected such that they are symmetrically equivalent with respect to a threefold axis. In all cases, a face of the `domain of influence' is perpendicular to each basis vector of these primitive cells.
Further classifications use reduction theory. There are different approaches to the reduction of quadratic forms in mathematics. The two most important in our context are
The investigations by Gruber (cf. Chapter 9.3 ) have shown the common root of both crystallographic approaches. As in Chapters 9.2 and 9.3 the Niggli reduction will be discussed in detail, we shall discuss the Delaunay reduction here.
We start with a lattice basis . This basis is extended by a vector All scalar products are considered. The reduction is performed minimizing the sum It can be shown that this sum can be reduced as long as one of the scalar products is still positive. If e.g. the scalar product is still positive, a transformation can be performed such that the sum of the transformed is smaller than : In the twodimensional case, holds.
If all the scalar products are less than or equal to zero, the three shortest vectors forming the reduced basis are contained in the set which corresponds to the maximal set of faces of the Dirichlet domain (14 faces).
For practical application, it is useful to classify the patterns of the resulting scalar products regarding their equivalence or zero values. These classes of patterns correspond to the reduced bases and result in `Symmetrische Sorten' (Delaunay, 1933) that lead directly to the conventional crystallographic cells by fixed transformations (cf. Patterson & Love, 1957; Burzlaff & Zimmermann, 1993). Table 9.1.8.1 gives the list of the 24 `symmetrische Sorten'. Column 1 contains Delaunay's symbols, column 2 the symbol of the Bravais type. For monoclinic centred lattices, the matrix of the last column transforms the primitive reduced cell into an Icentred cell, which has to be transformed to A or C according to the monoclinic standardization rules, if necessary. operates on the basis in the following form: where denotes the matrix of the basis vectors of the reduced cell and the matrix of the conventional cell.

Column 3 gives metrical conditions for the occurrence of certain Voronoi types. Column 5 indicates the relations among the scalar products of the reduced vector set. In some cases, different Selling patterns are given for one `symmetrische Sorte'. This procedure avoids a final reduction step (cf. Patterson & Love, 1957) and simplifies the computational treatment significantly. The number of `symmetrische Sorten', and thus the number of transformations which have to be applied, is smaller than the number of lattice characters according to Niggli. Note that the introduction of reduced bases using shortest lattice vectors causes complications in more than three dimensions (cf. Schwarzenberger, 1980).
This example is discussed in Azároff & Buerger (1958, pp. 176–180).
The lattice parameters are given as , , , , , . The scalar products resulting from these data are given in Table 9.1.9.1. The scalar product is positive. Thus the transformation is applied. The new scalar products are all nonpositive as given in the second row of Table 9.1.9.1 (within the accuracy of the experimental data). Comparison with Table 9.1.8.1 leads to M6, Voronoi type IV and the monoclinic Bravais lattice mP.

The transformation related to this case leads to a monoclinic conventional cell but does not consider the possibility of shorter basis vectors. For this reason, it is necessary here to look at the other vectors of the set V in the () plane, the only one of interest is . The length of this vector is , which is shorter than () and leads to the cell parameters .
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